Balancing Graphs in O(n*log(n))

The Barnes-Hut algorithm is used to model N-body systems more efficiently, at the expense of precision. However, the improvement in performance is tremendous: from O( n² ) to O( n*log(n) ). For reference, this is what that looks like:

Since our force-based approach to computing graph layout is essentially nothing other than an N-body simulation, and we don't require great precision anyway, this algorithm seems perfect for us to use. The way it works is basically this:

• The set of nodes is sorted into a quadtree, which is essentially a binary tree, but with an added dimension
• Each of the quadtree sectors keeps track of its mass and center of mass
• To compute the forces, the tree is traversed and for points that are "far" from a certain center of mass, only the force from the node to that center of mass is calculated, as opposed to each force between all nodes

This leads to a significant decrease in overall calculations that needs to be done, since distant nodes no longer need to interact directly. There is some overhead on creating the quadtree every frame, but the growth of this is also O(n*log(n)). If you want to learn more, here is a very good explanation of the algorithm complete with pseudocode (in German, but the code is understandable anyway). 

I haven't had time to implement this algorithm in our Qt application to graph Wikipedia, however I managed to code up the basic algorithm in Python. The improvement is tremendous, before I was never able to reasonably run a simulation with more than 80 or so nodes in Python, now even systems up to 350 nodes run at a reasonable framerate.

Illustration of the quadtree (unbalanced)

The same graph after balancing

The algorithm can also be parallelized very well, so the potential of this is tremendous, with good implementation and appropriate hardware we could realistically graph all of the German Wikipedia (~1.3 million articles), maybe more. 

The full Python code is here.

9868 nodes in WikiGraph

Ha! I located one of the bugs, apparently the setting of the repulsion strength parameter in the balancing algorithm caused the flying off into infinity of nodes. I fixed it, and right away managed to graph almost 10,000 articles centered on Mathematics! I could probably go larger, but each of these Graphs is taking up to half an hour to fully balance on my computer. Let's see, maybe TUM has a nice multicore machine we can use for a bit...

A little Preview

So, there haven't been many posts on here the last few days, but certainly not because I've gotten lazy. Robert and I have been coding like crazy on the WikiGraph project this week, with a few sessions of 6+ hours at times.
We're coming along quite well, we've managed to implement multithreading on all of the force calculations, which has given us a speedup factor of 4 on my machine, but the implementaton allows for an arbitrary number of threads and automatically creates as many as the machine it runs on has cores (or virtual cores, with hyperthreading). Along with that comes the improved algorithm, which again about doubles performance.
This allows us to display some really large graphs - by really large, I mean up to around 6000 nodes at the moment, and that limit is mostly due to some unexpected behavior (nodes flying off to infinity), which we suspect is a result of the parameters we use in our algorithm. I expect that by the time we're releasing a beta version (which at current development speed would be sometime next week), we can display graphs of maybe 15000 nodes, which is around 3 levels deep of pagelinks if all links are loaded.
OpenGL support is also something we're implementing at the moment, since CPU rendering of these graphs actually takes up more resources and time than calculating a single iteration of forces, so outsourcing that to a graphics card will definitely help performance.
Apart from that, other changes will mostly be in UI and design, adding options for users to alter the graph and such.

Even though we aren't ready to put out a stable and usable release yet, we've already created some pretty neat visualizations:

Close up of an earlier version, 2 levels centered around Quicksort

3 levels deep (German Wikipedia), also centerd around Quicksort I believe. We're actually displaying over 5000 nodes here.

3 levels around Mathematics, over 6000 nodes. This one is not quite stable, the Mathematics article needs to be fixed otherwise it flies off. Looking into why it does that, all other nodes seem to behave as expected.

An improved algorithm for N-body simulations

TL;DR: A simple algorithm that speeds up force calculations for n-body systems by a factor of nearly 2.

So the last post on here was about graph balancing using a spring-electrical model, in which each node exerts a repulsion on each other node according to Coulomb's law, and each edge attracts adjacent nodes according to the spring equation.
This is a good example of an N-body problem - the most well-known one is that of planetary/gravitational systems - and one unfortunate thing about these is that modeling them can be quite slow.
Intuitively, the first algorithm that comes to mind is simply iterating through all the bodies, and for each one iterate through each other body and calculating the force they exert on each other. For n bodies (nodes), this results in (n-1)^2 for the whole system - (n-1) because the body does not exert a force on itself. This is of course is a growth rate of O(n^2), not very efficient.
But there's an easy improvement that can improve on this quite a bit making only minimal changes to the algorithm:

• We start with the first body, and iterate through all the other body, calculating each force and adding it to the nodes net force
• Since the forces between each two bodies will be equal and opposite, we now subtract this force (or add it negatively) from the other bodies' net forces, and then add a reference to the original body to a list (or in Qt a QSet, which is faster) that keeps track of all the bodies that have already been calculated
• We go to the next body, and again iterate through all other bodies, except now we skip any bodies already in our list of calculated bodies. This results in one less calculation being performed with each body we iterate through

And that's about it. With each node, we have one less calculation to perform, so our total number of operations is now:

This is significantly better, in fact:

 

For very large systems, this algorithm approaches to be twice as fast.

I implemented this algorithm in our Wikipedia grapher, and it's working quite nicely. Here's one of the graphs I made with it today(554 nodes, zoomed out):

There are also n log(n) algorithms for these systems, but frankly I don't understand how they work well enough to write about or implement one yet, so I'll call it a night for now and get to those another time.